Question

An airplane is preparing to land at an airport. It is 59,400 feet above the ground and is descending at the rate of 3,000 feet per minute. At the same airport, another airplane is taking off and will ascend at the rate of 2,400 feet per minute. When will the two airplanes be at the same altitude and what will that altitude be? Use pencil and paper. Use two other methods to solve the problem. Explain which methods are easier to use and which are more difficult to use for the situation.

Product: Your report needs to include the following:

a. Define the variables that you will be using.

b. Create a system of equations.

c. Solve the system using graphing. Be sure to include the graph in your report.

d. Solve the system using either substitution or elimination to check your graphing work.

Explain what your solution means in the context of the problem.

Answer 1

a. Variables:

Let's define the variables for the problem:

t: time in minutes (the variable we are solving for)

h1: altitude of the descending airplane in feet

h2: altitude of the ascending airplane in feet

b. System of Equations:

Based on the given information, we can create a system of equations:

Equation 1: h1 = 59,400 - 3,000t (altitude of descending airplane)

Equation 2: h2 = 0 + 2,400t (altitude of ascending airplane)

c. Solving the System using Graphing:

To solve the system using graphing, we need to plot the two equations on a graph and find the point of intersection. Here's the graph:

[Graph]

From the graph, we can see that the two lines intersect at a point. The x-coordinate of the point of intersection represents the time (t), and the y-coordinate represents the altitude (h) where the two airplanes are at the same altitude.

d. Solving the System using Substitution or Elimination:

To check our graphing work, we can solve the system using either substitution or elimination. Let's use substitution:

From Equation 1: h1 = 59,400 - 3,000t

Substitute h1 into Equation 2:

59,400 - 3,000t = 2,400t

Add 3,000t to both sides:

59,400 = 5,400t

Divide both sides by 5,400:

t = 11

Now, substitute the value of t back into either Equation 1 or Equation 2 to find the altitude (h) at that time:

h1 = 59,400 - 3,000 * 11

h1 = 59,400 - 33,000

h1 = 26,400 feet

Therefore, the two airplanes will be at the same altitude of 26,400 feet after approximately 11 minutes.

Explanation of Methods:

Graphing: Graphing the equations provides a visual representation of the problem and helps identify the point of intersection. However, it may not provide an exact value, and the precision depends on the accuracy of the graph.

Substitution or Elimination: These algebraic methods allow for precise calculations and the exact determination of the time and altitude at which the airplanes meet. They require more steps and calculations but offer a more accurate solution.

In the context of the problem, the solution means that after approximately 11 minutes, both airplanes will be at the altitude of 26,400 feet. At this moment, the descending airplane will have descended to the same altitude as the ascending airplane is ascending, resulting in the two airplanes being at the same height above the ground.

Explanation: